My wife is trying to do the question below. I cannot help out and wondered if someone could take me through it.
verifying that -1 is the identity is rather easy one you "know" -1 is the identity. but how did Plato know it was?
it involves a bit of "detective work". let's find out what the identity would have to be, if there actually is one.
by definition, an identity for * is an element e of Z, with:
a*e = e*a = a, for any element a of Z.
since we know * is commutative (because of Plato's post above), it suffices to find e with a*e = a, for all a (in Z).
so we write down an equation in Z, and "solve for e".
if a*e = a, what this means in terms of the operations we're used to is:
a+e+1 = a.
we can, once we're "back in the ordinary integers", subtract a from both sides, to get:
e+1 = 0, which makes it clear e = -1 (since that's the only integer e with e+1 = 0, or: if you like, subtract 1 from both sides).
we can use "the same trick" to find out what a-1 has to be.
by definition, a-1 is an element b of Z with a*b = b*a = e.
now, we know what "e" is (see above), so we do pretty much what we did above: write a*b = e in terms of regular operations in Z, and "solve for b".
a*b = e translates into:
a+b+1 = -1
subtract 1 from both sides:
a+b = -2
subtract a from both sides:
b = -2-a (since this formula only depends on "a", we're good).
my point being, it's often not profitable to just "guess" what the identity and/or inverse might be. often, with just a little bit of reasoning, we can figure out what they HAVE to be.